Sol. Let g and g` be the gravitational accelerations at the pole and at the equator respectively and consider a body of mass m on the surface of the planet, which has a mass M. at the pole,

mg = ,

giving GM = gR 2 .

At the equator, we have

= – mg` = mg – = .

Hence g = 2V 2 /R.

If we define gravitational potential energy with respect to a point at infinity from the planet, the body will have potential energy

.

Note that the negative sign in front of the gravitational force takes account of its attractiveness. The body at the pole then has total energy

E = mV 2 – .

For it to escape from the planet, its total energy must be at least equal to the minimum energy of a body at infinity, i.e. zero. Hence the escape velocity v is given by

mv 2 – = 0

or v 2 = = 2gR = 4V 2 ,

i.e. v = 2V.